Aeroelastic instability of paper sheet in an offset printing press

Abstract

A simplified and efficient computational model of a thin flexible plate coupled to one-sided axial narrow-channel airflow is presented, modeling flow-induced vibration of a paper sheet transported by short pulses of compressed air encountered in the infeed system of offset printing press machines. The mathematical model is based on a discrete chain of masses, torsional springs, dampers and massless rigid rods. The linearized equations of motion coupled to inviscid flow equations are derived and solved analytically in frequency domain. The structural model is validated on the case of bending of a clamped-free beam under gravity and the elastic and damping constants are identified from measurements. Aeroelastic instability is investigated for three fluid-to-solid mass ratios ranging between 0.24 and 1.4, corresponding to three types of paper used in real offset printing machines. The growth rates and frequencies of the coupled system for supercritical flow velocities are reported and analyzed.

Appendix A: Derivation of the aerodynamic forces

The mean velocity $U_0(x)$ is assumed zero upstream of the nozzle, and constant downstream:

$$\begin{aligned} U_0(x) = \left\{ \begin{matrix} 0 &{}\quad x < L_1 \\ U_0 &{}\quad x \ge L_1 \end{matrix} \right. \end{aligned}$$

(A.1)

The velocity perturbation derivative can be expressed from the linearized continuity equation (14) as

$$\begin{aligned} \frac{\partial {\tilde{u}}}{\partial x} = - \frac{U_0}{H_0} \frac{\partial w}{\partial x} - \frac{1}{H_0} \frac{\partial w}{\partial t}. \end{aligned}$$

(A.2)

In the interval downstream of the nozzle, $x \in \left<L_1,L \right>$, integration of (A.2) yields

$$\begin{aligned} \int _{L_1}^x \frac{\partial {\tilde{u}}}{\partial x} \mathrm{d}\xi = {\tilde{u}}(x) - {\tilde{u}}(L_1) = - \frac{U_0}{H_0} \left[ w(x,t) - w(L_1,t) \right] - \frac{1}{H_0} \int _{L_1}^x \frac{\partial w}{\partial t}(\xi ,t) \mathrm{d}\xi . \end{aligned}$$

(A.3)

Using Eq. (10) and boundary condition ${\tilde{u}}(L_1)=0$, the velocity perturbation can be written as

$$\begin{aligned} {\tilde{u}}(x,t) = - \frac{U_0}{H_0} \left[ \sum _{j=1}^{n-1} w_j(t) f_j(x) - w_k(t) \right] - \frac{1}{H_0} \sum _{j=1}^{n-1} {\dot{w}}_j(t) [ g_j(x) - g_j(L_1) ] \quad \forall x \in \left<L_1,L \right>, \end{aligned}$$

(A.4)

where

$$\begin{aligned} g_j(x) = \int _0^x f_j(\xi ) \ \mathrm{d} \xi , \quad h_j(x) = \int _0^x g_j(\xi ) \ \mathrm{d} \xi , \quad s_j(x) = \int _0^x h_j(\xi ) \ \mathrm{d} \xi \end{aligned}$$

(A.5)

are integrals of the Heaviside lambda function which can be precomputed analytically ($h_j$ and $s_j$ will be used later). After integration of the linearized Euler equation (13), one gets

$$\begin{aligned} \int _x^L \frac{\partial p}{\partial x} = p(L,t)-p(x,t) = - \rho \int _x^L \frac{\partial {\tilde{u}}}{\partial t}(\xi ,t) \mathrm{d}\xi - \rho U_0 [{\tilde{u}}(L)-{\tilde{u}}(x)] \quad \forall x \in \left<L_1,L \right>. \end{aligned}$$

(A.6)

Combining (A.4) with (A.5) and using boundary condition $p(L,t)=0$, the perturbation pressure takes the following form:

$$\begin{aligned} p(x,t)= & {} \frac{2 \rho U_0}{H_0} \sum _{j=1}^{n-1} {\dot{w}}_j(t) [g_j(x)-g_j(L)] + \frac{\rho U_0}{H_0} {\dot{w}}_k(t) (L-x) \nonumber \\&+ \frac{\rho }{H_0} \sum _{j=1}^{n-1} \ddot{w}_j(t) \left[ h_j(x)-h_j(L)+g_j(L_1)(L-x) \right] \nonumber \\&+ \frac{\rho U_0^2}{H_0} \sum _{j=1}^{n-1} w_j(t) [f_j(x) - f_j(L)] \quad \forall x \in \left<L_1,L \right>. \end{aligned}$$

(A.7)

A similar procedure can be performed in the interval upstream of the nozzle, where the boundary conditions are ${\tilde{u}}(L_1)=0, p(0,t)=0$. Eventually,

$$\begin{aligned} {\tilde{u}}(x,t)= & {} \frac{1}{H_0} \sum _{j=1}^{n-1} {\dot{w}}_j(t) [ g_j(L_1) - g_j(x) ] \quad \forall x \in \left<0,L_1\right>, \end{aligned}$$

(A.8)

$$\begin{aligned} p(x,t)= & {} - \frac{\rho }{H_0} \sum _{j=1}^{n-1} \ddot{w}_j(t) \left[ g_j(L_1)x - h_j(x) \right] \quad \forall x \in \left<0,L_1\right>. \end{aligned}$$

(A.9)

The discrete aerodynamic forces $F_i(t)$ at the locations of mass with indices i can be then expressed as

$$\begin{aligned} F_i(t) = b \int _{x_i - \frac{\varDelta }{2}}^{x_i+ \frac{\varDelta }{2}} p(x,t) \ \mathrm{d}x = \rho \sum _{j=1}^{n-1} a_{ij} \ddot{w}_j + \rho U_0 \sum _{j=1}^{n-1} b_{ij} {\dot{w}}_j + \rho U_0^2 \sum _{j=1}^{n-1} c_{ij} w_j. \end{aligned}$$

(A.10)

Coefficients $a_{ij}$, $b_{ij}$ and $c_{ij}$ in (A.10), which are constant and can be precomputed analytically, have the following form:

$$\begin{aligned} a_{ij}= & {} \left\{ \begin{array}{ll} - \frac{b}{H_0} \left[ g_j(L_1) \varDelta x_i - s_j(x_i+\frac{\varDelta }{2})+s_j(x_i-\frac{\varDelta }{2}) \right] &{}\quad i \in \left<0,k-1\right> \\ {\frac{b}{H_0} \left[ -h_j(L) \frac{\varDelta }{2} +g_j(L_1) \frac{\varDelta }{2} (L-\frac{\varDelta }{2}) + s_j(L_1+\frac{\varDelta }{2}) - 2 s_j(L_1) + s_j(L_1-\frac{\varDelta }{2}) \right] } &{}\quad i=k \\ \frac{b}{H_0} \left[ -h_j(L) \varDelta + g_j(L_1) \varDelta (L-x_i) + s_j(x_i+\frac{\varDelta }{2}) - s_j(x_i - \frac{\varDelta }{2}) \right] &{}\quad i \in \left<k+1,n\right> \end{array} \right. \end{aligned}$$

(A.11)

$$\begin{aligned} b_{ij}= & {} \left\{ \begin{array}{ll} 0 &{}\quad i \in \left<0,k-1\right> \\ \frac{2 b}{H_0} \left[ -g_j(L) \frac{\varDelta }{2} +h_j(L_1 + \frac{\varDelta }{2}) - h_j(L_1) \right] + \frac{b}{H_0} \frac{\varDelta }{2} \left( L - L_1 - \frac{\varDelta }{4} \right) \delta _{jk} &{}\quad i=k \\ \frac{2 b}{H_0} \left[ -g_j(L) \varDelta + h_j(x_i + \frac{\varDelta }{2}) - h_j(x_i - \frac{\varDelta }{2}) \right] + \frac{b}{H_0} \varDelta \left( L - x_i \right) \delta _{jk} &{}\quad i \in \left<k+1,n\right> \end{array} \right. \end{aligned}$$

(A.12)

$$\begin{aligned} c_{ij}= & {} \left\{ \begin{array}{ll} 0 &{}\quad i \in \left<0,k-1\right> \\ \frac{b}{H_0} \left[ -f_j(L) \frac{\varDelta }{2} + g_j(L_1) - g_j(L_1 - \frac{\varDelta }{2}) \right] &{}\quad i=k \\ \frac{b}{H_0} \left[ -f_j(L) \varDelta + g_j(x_i + \frac{\varDelta }{2}) - g_j(x_i - \frac{\varDelta }{2}) \right] &{}\quad i \in \left<k+1,n\right> \end{array} \right. \end{aligned}$$

(A.13)

where $\delta _{jk}$ is the Kronecker delta.